渗流驱动问题间断有限元高效数值方法研究
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摘要
间断有限元方法能够应用于流体力学计算领域,并受到人们的广泛关注,主要在于它具有以下优点:由于近似解的间断性假设,对网格正则性要求不高,不需要考虑像一般有限元方法中连续性的限制条件就可以对网格进行加密或减疏处理,而且不同的单元可以采用不同形式、不同次数的逼近多项式;它是局部质量守恒的;它几乎没有数值扩散,能够灵活处理数值解含有大梯度大变形和间断的问题,可以准确地捕捉到激波(或大梯度变化区域)的位置,有效地避免数值振荡;间断有限元方法比协调有限元更容易实现hp自适应;而且,在恰当的网格上,改变逼近空间多项式的次数p可以获得按指数收敛的数值解。
虽然有很多学者致力于渗流驱动问题数值方法的研究,但关于用间断有限元方法求解这类问题的较少。对于多孔介质渗流驱动问题(包括可压流和不可压流),我们采用间断有限元方法对空间进行离散求解。
对于不可压的情形,S.Sun和M.F.Wheeler等人作了大量工作,他们建立了Oden-Babu(?)ka-Baumann DG(OBB-DG),non-symmetric interior penaltyGalerkin(NIPG),symmetric interior penalty Galerkin(SIPG)和incomplete interiorpenalty Galerkin(IIPG)四种间断有限元方法半离散形式的能量模误差估计和SIPG的L~2(L~2)模和负模误差估计。但他们并没有用到对偶技巧。我们利用对偶论证,建立了上述四种半离散间断有限元方法的L~2(L~2)模一致后验误差估计式,对全离散格式也进行了分析和研究。
H.Chen和M.R.Cui对于可压缩渗流驱动问题的混合有限元/间断有限元耦合格式做了hp误差分析。但他们的分析仅只限于没有弥散的特殊情况:实际中,分子扩散和弥散的影响都是存在的,这给分析带来了一定的困难。对同时具有分子扩散和弥散效应的可压缩流方程和反应输运方程的耦合问题,我们采用间断有限元方法求解,运用归纳假设导出了先验hp误差估计,运用对偶论证导出了后验hp误差估计;也对于混合有限元/间断有限元耦合格式求解的思想作了论证,导出了先验和后验hp误差估计。在对可压缩渗流驱动问题的先验误差分析过程中,我们采用归纳假设,不象S.Sun所用的引入剪切算子,这样避免了选择该算子中那个大的正常数。在混合有限元方法的超收敛性方面国内外有大量的工作,但对间断有限元方法超收敛性的研究较少,尤其是针对渗流驱动问题的超收敛研究几乎没有。对同时具有分子扩散和弥散效应的不可压缩和可压缩渗流驱动问题进行了混合有限元/间断有限元耦合格式的超收敛分析研究。
Discontinuous Galerkin methods are very attractive for numerical simulations in computational fluid dynamics because of their physical and numerical properties. Firstly,it is flexible that allows for general non-conforming meshes with variable degrees of approximation and of higher accuracy.Secondly,it is locally mass conservative.Thirdly,it has less numerical diffusion and can also handle rough and discontinuous coefficient problems.Fourthly,it is easier for hp-adaptivity than conforming approaches because the information over cell boundaries is almost decoupled. In addition,with appropriate meshing,varying p can yield exponential convergence rates.
Incompressible and compressible miscible displacement problems in porous media are investigated.Several discontinuous Galerkin(DG) finite element methods are used for spatial discretization.For the incompressible case,many scholars such as S.Sun and M.F.Wheeler,have done a lot of work.They established the error estimates in a energy norm for four semi-discrete primal discontinuous Galerkin methods,i.e.,Oden-Babu(?)ka-Baumann DG(OBB-DG),non-symmetric interior penalty Galerkin(NIPG),symmetric interior penalty Galerkin(SIPG), and incomplete interior penalty Galerkin(IIPG) without duality assumption.The estimates in L~2(L~2) norm and in negative norm are also derived for SIPG by them. Based on the duality argument,we work out a unified a posteriori error estimate in L~2(L~2) norm for four semi-discrete and full-discrete primal DG methods above. H.Chen and M.R.Cui present hp error analysis for a combined mixed and discontinuous Galerkin method(MFE/DG) for compressible miscible displacement problems.But only the dispersion-free case is considered.In practical,the effects of molecular diffusion and dispersion are included,which makes our analysis more complicate and difficult.Based on the work of H.Chert and M.R.Cui,we consider the complete compressible case with no restrictions on the diffusion/dispersion tensor. Several DG approximations and combined MFE/DG methods are applied.A priori error estimates and a posteriori one are obtained using the induction hypothesis instead of the cut-off operator and duality technique,respectively.Finally, some superconvergence analysis of combined MFE/DG methods are investigated for incompressible and compressible miscible displacement problems.
虽然有很多学者致力于渗流驱动问题数值方法的研究,但关于用间断有限元方法求解这类问题的较少。对于多孔介质渗流驱动问题(包括可压流和不可压流),我们采用间断有限元方法对空间进行离散求解。
对于不可压的情形,S.Sun和M.F.Wheeler等人作了大量工作,他们建立了Oden-Babu(?)ka-Baumann DG(OBB-DG),non-symmetric interior penaltyGalerkin(NIPG),symmetric interior penalty Galerkin(SIPG)和incomplete interiorpenalty Galerkin(IIPG)四种间断有限元方法半离散形式的能量模误差估计和SIPG的L~2(L~2)模和负模误差估计。但他们并没有用到对偶技巧。我们利用对偶论证,建立了上述四种半离散间断有限元方法的L~2(L~2)模一致后验误差估计式,对全离散格式也进行了分析和研究。
H.Chen和M.R.Cui对于可压缩渗流驱动问题的混合有限元/间断有限元耦合格式做了hp误差分析。但他们的分析仅只限于没有弥散的特殊情况:实际中,分子扩散和弥散的影响都是存在的,这给分析带来了一定的困难。对同时具有分子扩散和弥散效应的可压缩流方程和反应输运方程的耦合问题,我们采用间断有限元方法求解,运用归纳假设导出了先验hp误差估计,运用对偶论证导出了后验hp误差估计;也对于混合有限元/间断有限元耦合格式求解的思想作了论证,导出了先验和后验hp误差估计。在对可压缩渗流驱动问题的先验误差分析过程中,我们采用归纳假设,不象S.Sun所用的引入剪切算子,这样避免了选择该算子中那个大的正常数。在混合有限元方法的超收敛性方面国内外有大量的工作,但对间断有限元方法超收敛性的研究较少,尤其是针对渗流驱动问题的超收敛研究几乎没有。对同时具有分子扩散和弥散效应的不可压缩和可压缩渗流驱动问题进行了混合有限元/间断有限元耦合格式的超收敛分析研究。
Discontinuous Galerkin methods are very attractive for numerical simulations in computational fluid dynamics because of their physical and numerical properties. Firstly,it is flexible that allows for general non-conforming meshes with variable degrees of approximation and of higher accuracy.Secondly,it is locally mass conservative.Thirdly,it has less numerical diffusion and can also handle rough and discontinuous coefficient problems.Fourthly,it is easier for hp-adaptivity than conforming approaches because the information over cell boundaries is almost decoupled. In addition,with appropriate meshing,varying p can yield exponential convergence rates.
Incompressible and compressible miscible displacement problems in porous media are investigated.Several discontinuous Galerkin(DG) finite element methods are used for spatial discretization.For the incompressible case,many scholars such as S.Sun and M.F.Wheeler,have done a lot of work.They established the error estimates in a energy norm for four semi-discrete primal discontinuous Galerkin methods,i.e.,Oden-Babu(?)ka-Baumann DG(OBB-DG),non-symmetric interior penalty Galerkin(NIPG),symmetric interior penalty Galerkin(SIPG), and incomplete interior penalty Galerkin(IIPG) without duality assumption.The estimates in L~2(L~2) norm and in negative norm are also derived for SIPG by them. Based on the duality argument,we work out a unified a posteriori error estimate in L~2(L~2) norm for four semi-discrete and full-discrete primal DG methods above. H.Chen and M.R.Cui present hp error analysis for a combined mixed and discontinuous Galerkin method(MFE/DG) for compressible miscible displacement problems.But only the dispersion-free case is considered.In practical,the effects of molecular diffusion and dispersion are included,which makes our analysis more complicate and difficult.Based on the work of H.Chert and M.R.Cui,we consider the complete compressible case with no restrictions on the diffusion/dispersion tensor. Several DG approximations and combined MFE/DG methods are applied.A priori error estimates and a posteriori one are obtained using the induction hypothesis instead of the cut-off operator and duality technique,respectively.Finally, some superconvergence analysis of combined MFE/DG methods are investigated for incompressible and compressible miscible displacement problems.
引文
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